Decoherence during inflation: The generation of classical inhomogeneities

We show how the quantum-to-classical transition of the cosmological fluctuations produced during inflation can be described by means of the influence functional and the master equation. We split the inflaton field into the system field (long-wavelength modes), and the environment, represented by its own short-wavelength modes. We compute the decoherence times for the system-field modes and compare them with the other time scales of the model. We present the renormalized stochastic Langevin equation for an homogeneous system field and then we analyze the influence of the environment on the power spectrum for some modes in the system.

Figure 1

Coefficient $D_1$ (in logarithmic scale) as a function of $k_0$ for $H\eta =-0.5$.

Figure 2

Coefficient $D_1$ (in logarithmic scale) as a function of $k_0$ for a fixed value of $|k_0\eta |=1/10$ ($k_0>H/10$), where we have also plotted the approximation in Eq. (33).

Figure 3

Temporal integral of $D_1$ in units of $H^{-1}$ as a function of $\mathcal{N}[k_0,{\eta }_d]$ for two different values of $k_0$. Solid curves are numerical integrations of the exact coefficient, while short-dashed curves are from analytical integration of Eq. (33). Note the logarithmic scale on the vertical axis.

Figure 4

Coefficient $D_2$ (in logarithmic scale) as a function of $\Lambda /H$ for a fixed conformal time $\eta =-{10}^{-3}{H}^{-1}$ and three different values of $k_0$ ($H/10$, $H$, and $100H$).

Figure 5

Coefficient $D_2$ as a function of $k_0$ for $\Lambda =20H$ and $H\eta =-1/2$.

Figure 6

The same as Fig. 5 but for modes $k_0$ outside the Hubble radius at $\eta =-1/2H^{-1}$ and $\Lambda =2H$.

Figure 7

Coefficient $D_2$ (in logarithmic scale) as a function of $\mathcal{N}[k_0,{\eta }_d]$ for $\Lambda =300H$ and $k_0=100H$. Heavy solid curve is numerically calculated from Eq. (27) (see details in Appendix ). Light solid curve corresponds to the approximation obtained by averaging the local (dotted curve) and the asymptotic (short-dashed curve) approximations.

Figure 8

$D_2$ and $D_2^{\mathrm{approx}}$ as functions of $k_0/H$ for a fixed value of $|k_0\eta |=1/10$ ($k_0>H/10$) and $\Lambda =100H$. The vertical axis is on a logarithmic scale.

Figure 9

Temporal integrals of $D_2$ and $D_2^{\mathrm{approx}}$ (on a logarithmic scale and in units of $H^{-1}$) as functions of $\mathcal{N}[k_0,{\eta }_d]$, for $\Lambda =300H$ and two different values of $k_0$ ($100H$ and $200H$).

Figure 10

The same as Fig. 9 but for smaller values of $k_0$ ($k_0=H$ and $k_0=H/10$) and $\Lambda =2H$.

Figure 11

$\mathcal{J}$ and ${\mathcal{J}}_{1,2,3}$ as functions of $\mathcal{N}[\Lambda ,{\eta }^{\prime }]\equiv -\ln |\Lambda {\eta }^{\prime }|$ for $\Lambda \eta =-{10}^{-2}$ ($\mathcal{N}[\Lambda ,\eta ]=\ln 100\simeq 4.605$).

Figure 12

$\mathcal{R}$ as a function of $\mathcal{N}[\Lambda ,{\eta }^{\prime }]$ for $\Lambda \eta =-{10}^{-2}$ ($\mathcal{N}[\Lambda ,\eta ]=\ln 100\simeq 4.605$).

Figure 13

${\Delta }_{{\varphi }^{\xi }}^2/(\lambda {\varphi }_0{)}^2$ as a function of $\mathcal{N}[k_0,\eta ]$ for $\Lambda =5H$ and two different values of $k_0$. Note that the vertical axis is on a logarithmic scale.

Figure 14

${\Delta }_{{\varphi }^{\xi }}^2/(\lambda {\varphi }_0{)}^2$ as a function of $k_0/H$ for $\Lambda =5H$ and $|k_0\eta |=0.001$.

Figure 15

${\Delta }_{{\varphi }^{\xi }}^2/(\lambda {\varphi }_0{)}^2$ as a function of $\Lambda /H$ for $k_0=2H$ and $|k_0\eta |=0.001$.